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RELFE BROTHERS' EUCLID SHEETS,

PROPOSITIONS 1—26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.

PROPOSITION VI.

If two angles of a triangle be equal to each other; the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

Let be a triangle having the angle equal to the angle
Then the side shall be equal to the side

9

For, if

be not equal to one of them is greater than the other. If possible, let be greater than ; and from cut off

equal to

the less, and join Then, in the triangles

because is equal to

and is common to both triangles, the two sides

are equal to the two sides

each to each ; and the angle is equal to the angle ; therefore the base
is equal to the base
and the triangle is equal to the triangle

the less equal to the greater, which is absurd. Therefore is not unequal to that is,

is equal to

Wherefore, if two angles, &c.

RELFE BROTHERS' EUCLID SHEETS.

PROPOSITIONS 1–26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS

PROPOSITION V.

The angles at the base of an isosceles triangle are equal to each other; and if the equal sides be produced, the angles on the other side of the base shall be equal.

; from

Let

be an isosceles triangle of which the side is equal to and let the equal sides

be produced to and Then the angle shall be equal to the angle and the angle to the angle In take any point

the greater, cut off

equal to the less, and join Because is equal to

and
to ; the two sides

are equal to the two each to each; and they contain the angle common to the two triangles

;

A

; and

therefore the base is equal to the base

and the triangle is equal to the triangle also the remaining angles of the one are equal to the remaining angles of the other, each to each, to which the equal sides are opposite; viz., the angle

to the angle

and the angle to the angle And because the whole is equal to the whole

of which the parts are equal; therefore the remainder is equal to the remainder

has been proved to be equal to ; hence, because the two sides

are equal to the two each to each; and the angle has been proved to be equal to the angle

also the base is common to the two triangles

; wherefore these triangles are equal, and their remaining angles, each to each, to which the equal sides are opposite; therefore the angle

is equal to the angle

and the angle

to the angle And, since it has been demonstrated, that the whole angle is equal to the whole the parts of which, the angles are also equal ; therefore the remaining angle

is equal to the remaining angle which are the angles at the base of the triangle ; and it has also been proved, that the angle is equal to the angle

which are the angles upon the other side of the base. Therefore the angles at the base, &c.

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RELFE BROTHERS' EUCLID SHEETS.
PROPOSITIONS 1—26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.

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If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to each other; they shall likewise have their bases or third sides equal, and the two triangles shall be equal, and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite. Let be two triangles, which have the two sides

equal to the two sides each to each, viz., to and

and the included angle equal to the included angle Then shall the base be equal to the base ; and the triangle

to the triangle ; and the other angles to which the equal sides are opposite shall be equal, each to each, viz., the angle

to the angle
and the angle

to the angle

be on

; and

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For, if the triangle be applied to the triangle , so that the point may

and the straight line on ; then the point shall coincide with the point, because is equal to

coinciding with the straight line shall fall on because the angle is equal to the angle

therefore also the point shall coincide with the point because is equal to ; but the point was shewn to coincide with the point ;

wherefore the base shall coincide with the base ; because the point

coinciding with and with if the base do not coincide with the base the two straight lines and would enclose a space, which is impossible.

Therefore the base does coincide with and is equal to it; and the whole triangle coincides with the whole triangle and is equal to it; also the remaining angles of one triangle coincide with the remaining angles of the other, and are equal to them, viz., the angle

to the angle
and the angle

to
Therefore, if two triangles have two sides of the one equal to two sides, &c.

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